The generalization to \(r\)  boundary components (where \(r \ge 2\) ) is given in
(5.6) of [1]}:
\( {\mathcal {N}}^{\mathfrak {g}, r} \,=\, \lbrace  (a_1,\, \dots ,\,a_g,\, b_1,\,\dots ,\, b_g,\, k_2,\, \dots ,\, k_r,\, \lambda _1,\,\dots ,\, \lambda _r )\ \Big \vert \ a_j,\, k_j \,\in \, G,\) 
\(\lambda _j \,\in \, \mathfrak {g},\prod _{i=1}^g [a_{i}, \,b_{i}]\, =\, e^{\lambda _1} k_2 e^{\lambda _2} k_2^{-1}\dots k_r e^{\lambda _r} k_r^{-1} \rbrace . \) 
